MATH 180  Precalculus5 Credit: (5 lecture, 1 lab, 0 clinical) 6 Contact Hours: [Math Level 6 ] This course is designed to provide students with a clear understanding of functions as a solid foundation for subsequent courses. Functions studied include linear, quadratic, exponential, logarithmic, trigonometric, polynomial, and rational. Other topics will include modeling, concavity, transformations of functions, compositions and combinations of functions, sequences, series and parametric equations. This course stresses conceptual understanding and multiple ways of representing mathematical ideas. OFFERED: spring semesters
Course Goals/ Objectives/ Competencies: Goal 1: Students will deepen their understanding of functions.
Objectives: The student will
 demonstrate an understanding of functions and function notation.
 interpret function notation.
 describe and analyze a function graphically, analytically, numerically, and verbally.
 evaluate functions and interpret the results graphically, analytically, numerically, and verbally.
 calculate and analyze the average rate of change of a function on an interval.
 given a function graphically, analytically, numerically, or verbally, find a value of the domain for a given value of the range and viceversa and understand the function notation associated with each scenario.
 use average rate of change to determine if a function is increasing or decreasing on an interval.
 describe the concavity of a function given graphically, analytically, numerically, or verbally.
 describe and analyze the relationship between the concavity of a function and the rate of change of the function.
 determine the domain and range of a function given graphically, analytically, numerically, or verbally.
Goal 2: Students will investigate piecewise functions, inverse functions, composition of functions and combinations of functions.
Objectives: The student will
 evaluate and graph piecewise defined functions.
 write the formula for a piecewise functions and use piecewise functions to model applications.
 determine the domain and range of a piecewise function.
 calculate the composition of functions graphically, analytically and numerically.
 write the formula for a composite function.
 use composite functions to model applications.
 decompose functions.
 use inverse function notation and interpret this notation in the context of a problem.
 find and interpret the formula for an inverse function.
 state the domain and range of a function and its inverse.
 distinguish between invertible and noninvertible functions.
 verify that two functions are inverses algebraically through composition.
 sketch and analyze the graphs of inverse functions.
 sketch the graph of the combination of functions
 evaluate the combination of functions.
Goal 3: Students will deepen their understanding of quadratic functions.
Objectives: The student will
 find and analyze the zeros of a quadratic function using several methods.
 use average rate of change to describe the concavity of a quadratic function.
 write the formula of a quadratic function in factored form given different characteristics of the function.
 rewrite and analyze the formula of a quadratic function in vertex form.
 use the vertex form of a quadratic function to determine if the function will have 0, 1, or 2 real zeros.
 set up and solve application problems involving quadratic functions.
 determine the nonreal zeros of a quadratic function.
 perform arithmetic calculations with complex numbers.
Goal 4: Students will investigate exponential functions.
Objectives: The student will
 describe and analyze the behavior of exponential functions graphically, analytically, numerically, and verbally.
 write and analyze formulas for exponential functions given graphically, numerically, or verbally.
 compare the characteristics, behaviors, formulas, graphs and applications of linear and exponential functions.
 graph exponential functions and compare the graphs of exponential functions.
 set up and solve application problems involving exponential functions.
 write and solve an exponential equation graphically.
 demonstrate an understanding of the applications of exponential functions to compound interest.
 identify and analyze the nominal and effective annual rates for compounded interest.
 write and analyze a formula for an exponential function with a base of e that is given graphically, numerically, or verbally.
 compare exponential functions of the form Q = ab^{t} and Q = ae^{kt }
Goal 5: Students will investigate logarithmic functions.
Objectives: The student will
 define log_{b}x.
 rewrite equations containing logarithms using exponents and vice versa.
 evaluate and simplify expressions with logarithms.
 use logarithms to solve exponential equations.
 set up and solve different application problems such as half life and doubling time with logarithms.
 rewrite a function of the form Q = ab^{t} in the form Q = ae^{kt }and vice versa.
 graph logarithmic functions and identify geometric qualities of the functions.
 state the domain and range for any logarithmic functions based on the definition of log_{b}x.
 apply and analyze logarithmic models.
 solve logarithmic equations.
 demonstrate an understanding of logarithmic scales.
 apply and interpret logarithmic scales to analyze data.
Goal 6: Students will investigate transformations of functions and their graphs.
Objectives: The student will
 sketch y = g(x) + k and y = g(x + k) given a table of values or the graph of y = g(x).
 write the formula for a function given graphically or numerically that has been shifted vertically and/or horizontally.
 identify and analyze the symmetry of a function given analytically, graphically, numerically or verbally.
 sketch y = kg(x) and y = g(kx) given a table of values or the graph of y = g(x).
 write the formula for a function given graphically or numerically that has been stretched or compressed vertically and/or horizontally.
 describe, graph and write a formula of a function given analytically, graphically, numerically or verbally that has multiple transformations.
 compare the effects of the order in which transformations are applied on the graphs and formulas of a transformed function.
Goal 7: Students will explore the idea of trigonometry in circles and triangle.
Objectives: The student will
 describe and analyze the relationship between the coordinates of a point on the unit circle and the sine and cosine of the corresponding angle.
 evaluate the sine and cosine of an angle given in a right triangle.
 sketch and analyze the graphs of basic sine and cosine functions.
 write and analyze the formula for a sine and cosine function that is given graphically, numerically, or verbally.
 evaluate the tangent of an angle using the unit circle.
 demonstrate an analytic understanding of the inverse trigonometric functions.
 apply the law of sines and the law of cosines.
 identify, describe and analyze the ambiguous case.
Goal 8: Students will become familiar with the trigonometric function.
Objectives: The student will
 define an angle that has a measure of 1 radian.
 find angles in radian measure given the number of revolutions around a circle or an arc length and the radius of the circle.
 convert between radian and degree measure.
 compare the evaluation of the sine and cosine of an angle in radian and degree measure.
 identify and interpret the period, midline and the amplitude of a sinusoidal function.
 write and analyze the formula for a sinusoidal function that is given graphically and numerically.
 describe the transformations of a sinusoidal function.
 graph the reciprocal trigonometric functions.
 solve trigonometric equations graphically and algebraically.
 evaluate the arcsine, arccosine or arctangent for a given value.
Goal 9: Students will investigate with the trigonometric identities.
Objectives: The student will
 describe the difference between an equation and an identity.
 apply the Pythagorean and double angle identities to rewrite and simplify expressions.
 solve trigonometric equations using the Pythagorean and double angle identities.
 apply the sum and difference formulas to rewrite and simplify expressions.
 verify and derive identities algebraically or graphically.
Goal 10: Students will study polynomial and rational functions.
Objectives: The student will
 translate “is proportioned to” into a mathematical equation.
 identify the form of a power function.
 describe geometric behavior of power functions graphically, analytically and verbally.
 find equations for power functions based on given data points.
 identify the form of a polynomial function.
 describe the characteristics of a polynomial function by observing the similarities to a power function.
 sketch the graph of a polynomial without a calculator if the zeros and degree of the polynomial are known.
 write a formula for a polynomial given its graph.
 identify the form of a rational function.
 describe geometric qualities of rational functions.
 determine the zeros and asymptotes of rational functions.
 write a formula for a rational function given its graph or table.
 compare the characteristics, behaviors, formulas, and applications of power, exponential, and logarithmic functions.
Goal 11: Students will explore sequences and series.
Objectives: The student will
 describe and analyze sequences.
 find a term for a sequence given a formula for the nth term.
 identify arithmetic and geometric sequences.
 write and apply the formula for the nth term of a arithmetic or geometric sequence.
 find and interpret the partial sum of arithmetic series or geometric series.
 write the formula for a sum in sigma notation.
 find and interpret the infinite sum of a geometric series.
 compare finite and infinite geometric series.
Goal 12: Students will study parametric equations.
Objectives: The student will
 graph and describe the motion given by parametric equations.
 eliminate the parameter from parametric equations.
 parameterize curves given analytically and graphically.
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